Common to all three types is the sequence of
compression space -- cooler --- regenerator --- heater -- expansion space
The alpha-type engine needs two pistons, compression and expansion piston, which change the volume of their respective adjacent spaces. Both pistons have to seal well against the ambient air. Often, the two pistons do not reside in the same cylinder.
Both, the beta and gamma type, need only a single piston which usually is the compression piston (always next to the cooler). The intent is to have this piston, which still has to seal well against the ambient air, operating at as low a temperature as possible. The expansion piston of the alpha type is replaced by the internal displacer piston. Its function is to move, at appropriate times, the working fluid back and forth between the expansion and compression space. The displacer does not have to seal quite as well as the power piston because it has to seal only against the pressure difference between the expansion and compression space. This pressure difference is usually very small because it is only due to the fluid friction in the cooler-regenerator-heater chain. Power piston and displacer reside inside the same cylinder for the beta-type and in separate cylinders for the gamma-type Stirling Engine.
A stark variation of the gamma-type engine are the LTD Stirling engines (Low Temperature Differential) which I call delta-type Stirling engines.
These engines are the favorite of many Stirling Engine enthusiasts and are known by names like coffee-cup, tin-can or pan-cake engines and can run on very low temperature differences between the hot and the cold side, ie run between the warmth of your hand and ambient air temperature. In most cases these engines dispense of the regenerator completely which reduces the already low power output of these engines even more. Heater and cooler are reduced to simple flat plates. Also note that there is an intentional gap between the displacer piston and the surrounging cylinder wall so that the working fluid can easily move from the hot to the cold side and back.